文档介绍：CONTENTS
Chapter I. Introduction 1
Chapter II. Modular symbol algorithms 7
Modular Symbols and Homology 7
The upper half-plane, the modular group and cusp forms 7
The duality between cusp forms and homology 9
Real structure 11
Modular symbol formalism 12
Rational structure and the Manin-Drinfeld Theorem 12
Triangulations and homology 13
M-symbols and Γ0(N) 15
Conversion between modular symbols and M-symbols 18
Action of Hecke and other operators 18
Working in H+(N) 23
Modular forms and modular elliptic curves 24
Splitting off one-dimensional eigenspaces 25
L(f, s) and the evaluation of L(f, 1)/Ω(f) 29
Fourier coefficients 31
periods I 33
periods II: Indirect method 37
periods III: Evaluation of the sums 41
L(r)(f, 1) 42
Obtaining equations for the curves 45
the degree of a modular parametrization 46
Modular Parametrizations 47
Coset representatives and Fundamental Domains 48
Implementation for Γ0(N) 50
Appendix to Chapter II. Examples 52
Example 1. N = 11 52
Example 2. N = 33 57
Example 3. N = 37 58
Example 4. N = 49 60
Chapter III. Elliptic curve algorithms 62
Terminology and notation 62
The Kraus–Laska–Connell algorithm and Tate’s algorithm 64
The Mordell–Weil group I: finding torsion points 68
Heights and the height pairing 71
The Mordell–Weil group II: generators 75
The Mordell–Weil group III: the rank 78
The period lattice 97
Finding isogenous curves 98
Twists plex multiplication 101
Chapter IV. The tables 104
Table 1. Elliptic curves 109
Table 2. Mordell–Weil generators 255
Table 3. Hecke eigenvalues 264
Table 4. Birch–Swinnerton-Dyer data 313
Table 5. Parametrization degrees 362
Bibliography 374
CHAPTER I
INTRODUCTION
Introduction to the First (1992) Edition
This book is in three section